Doubt continues to linger over the reality of quantum vacuum energy. There is some question whether fluctuating fields gravitate at all, or do so anomalously. Here we show that for the simple case of parallel conducting plates, the associated Casimir energy gravitates just as required by the equivalence principle, and that therefore the inertial and gravitational masses of a system possessing Casimir energy Ec are both Ec/c 2 . This simple result disproves recent claims in the literature. We clarify some pitfalls in the calculation that can lead to spurious dependences on coordinate system. PACS numbers: 03.70.+k, 04.20.Cv, 04.25.Nx, 03.65.Sq The subject of quantum vacuum energy (the Casimir effect) dates from the same year as the discovery of renormalized quantum electrodynamics, 1948 [1]. It puts the lie to the naive presumption that zero-point energy is not observable. On the other hand, it continues to be surrounded by controversy, in large part because sharp boundaries give rise to divergences in the local energy density near the surface (see Refs. [2,3,4]). The most troubling aspect of these divergences is in the coupling to gravity. Gravity has its source in the local energymomentum tensor, and such surface divergences promise serious difficulties. The gravitational implications of zero-point energy are an outstanding problem in view of our inability to understand the origin of the cosmological constant or dark energy [5,6,7].As a prolegomenon to studying such issues, we here address a simpler question: How does the completely finite Casimir energy of a pair of parallel conducting plates couple to gravity? The question turns out to be surprisingly less straightforward than one might suspect! Previous authors [8,9,10,11,12] have given disparate answers, including gravitational forces, or gravitationally modified Casimir forces, that depend on the orientation of the Casimir apparatus with respect to the gravitational field of the earth. We will here resolve some of this confusion with a convincingly calculated result consistent with the equivalence principle. That is, the renormalized Casimir energy couples to gravity just like any other energy. In our opinion, this fact is evidence that vacuum energy must be taken seriously in gravitational theory and that the problem of boundary divergences must be resolved by a better understanding of the modeling and renormalization processes.We start by recalling the electromagnetic Casimir stress tensor between a pair of parallel perfectly conducting plates separated by a distance a, with transverse dimensions L ≫ a, as given by Brown and Maclay [13]:where the third spatial direction is the direction normal to the plates. This is given in terms of the Casimir energy per unit area, E c = −π 2 c/(720a 3 ). Outside the plates, T µν = 0. Omitted here is a constant divergent term that is present both between and outside the plates, and also in the absence of plates, which cannot have any physical significance. Because the electromagnetic field respects conformal symmetr...

We derive boundary conditions for electromagnetic fields on a -function plate. The optical properties of such a plate are shown to necessarily be anisotropic in that they only depend on the transverse properties of the plate. We unambiguously obtain the boundary conditions for a perfectly conducting -function plate in the limit of infinite dielectric response. We show that a material does not ''optically vanish'' in the thin-plate limit. The thin-plate limit of a plasma slab of thickness d with plasma frequency ! 2 p ¼ p =d reduces to a -function plate for frequencies (( 1. We show that the Casimir interaction energy between two parallel perfectly conducting -function plates is the same as that for parallel perfectly conducting slabs. Similarly, we show that the interaction energy between an atom and a perfect electrically conducting -function plate is the usual Casimir-Polder energy, which is verified by considering the thin-plate limit of dielectric slabs. The ''thick'' and ''thin'' boundary conditions considered by Bordag are found to be identical in the sense that they lead to the same electromagnetic fields.

Motivated by a desire to understand quantum fluctuation energy densities and stress within a spatially varying dielectric medium, we examine the vacuum expectation value for the stress tensor of a scalar field with arbitrary conformal parameter, in the background of a given potential that depends on only one spatial coordinate. We regulate the expressions by incorporating a temporalspatial cutoff in the (imaginary) time and transverse-spatial directions. The divergences are captured by the zeroth-and second-order WKB approximations. Then the stress tensor is "renormalized" by omitting the terms that depend on the cutoff. The ambiguities that inevitably arise in this procedure are both duly noted and restricted by imposing certain physical conditions; one result is that the renormalized stress tensor exhibits the expected trace anomaly. The renormalized stress tensor exhibits no pressure anomaly, in that the principle of virtual work is satisfied for motions in a transverse direction. We then consider a potential that defines a wall, a one-dimensional potential that vanishes for z < 0 and rises like z α , α > 0, for z > 0. Previously, the stress tensor had been computed outside of the wall, whereas now we compute all components of the stress tensor in the interior of the wall. The full finite stress tensor is computed numerically for the two cases where explicit solutions to the differential equation are available, α = 1 and 2. The energy density exhibits an inverse linear divergence as the boundary is approached from the inside for a linear potential, and a logarithmic divergence for a quadratic potential. Finally, the interaction between two such walls is computed, and it is shown that the attractive Casimir pressure between the two walls also satisfies the principle of virtual work (i.e., the pressure equals the negative derivative of the energy with respect to the distance between the walls).

At air-water interfaces, the Lifshitz interaction by itself does not promote ice growth. On the contrary, we find that the Lifshitz force promotes the growth of an ice film, up to 1-8 nm thickness, near silica-water interfaces at the triple point of water. This is achieved in a system where the combined effect of the retardation and the zero frequency mode influences the short-range interactions at low temperatures, contrary to common understanding. Cancellation between the positive and negative contributions in the Lifshitz spectral function is reversed in silica with high porosity. Our results provide a model for how water freezes on glass and other surfaces. DOI: 10.1103/PhysRevB.95.155422 Although water in its different forms has been studied for a very long time, several properties of water and ice remain uncertain and are currently under intense investigation [1][2][3][4]. The question we want to address in the present paper is to what extent the fluctuation-induced Lifshitz interaction can promote the growth of ice films at water-solid interfaces, at the triple point of water. Particles and surfaces, e.g., quartz, soot, or bacteria, in supercooled water are known experimentally to nucleate ice formation [5][6][7]. Here, we focus on interfaces between water and silica-based materials and examine the roles of several intervening factors in the sum over frequency modes (Matsubara terms) contributing to the Lifshitz free energy.Quantum fluctuations in the electromagnetic field result in van der Waals interactions, which in their unretarded form were explained by London in terms of frequency-dependent responses to the fluctuations in the polarizable atoms constituting the material medium [8]. The understanding of these interactions was revolutionized when Casimir introduced retardation effects [9]. The theory was later generalized by Lifshitz to include dielectric materials [10,11]. The Lifshitz formula in Eq. (1), derived for three-layer planar geometries [11], gives the interaction energy between two semi-infinite dielectric media described by their frequency-dependent dielectric permittivities as well as the dielectric permittivity of the medium separating them (see Fig. 1).The purpose of the present work is twofold. First, we want to show that a finite size ice film, nucleated by a solid-water interface, can be energetically favorable even when only the Lifshitz interaction is accounted for. Second, we want to highlight a relevant contribution from the zero frequency term in the expression for the Lifshitz energy in a region where it is not expected to be important. The temperature dependence * Mathias.A.Bostrom@ntnu.no † prachi.parashar@ntnu.no ‡ iver.h.brevik@ntnu.no of the Casimir force between metal surfaces [11][12][13][14] relies strongly on the exact behavior of the low-frequency dielectric function of metals. These and many other investigations have provided support for the notion that the zero frequency term would only be relevant at high temperatures or large surface separations at a moderate ...